Pub Date : 2026-01-13DOI: 10.1016/j.fss.2026.109775
James C. Bezdek, Thomas A. Runkler
Possibilistic c-means (PCM) clustering began in 1993, and has been used since then in many applications. In this article we discuss the geometric foundations of PCM and introduce a new hard possibilistic c-means (HPCM) clustering algorithm. We use limit theory to prove that the extended set of possibilistic c-partitions is the unit hypercube in; and that its vertices are exactly the hard possibilistic c-partitions on n objects defined herein. This enables completion of the geometric description of the domain of possibilistic clustering algorithms. We give examples that compare the results of clustering with Hard c-means (HCM) to HPCM on three small synthetic data sets. Our proof-of-concept examples show that the new algorithm performs as expected, and provides much more realistic interpretation of clusters than HCM when the data contain bridge points or noise.
{"title":"Geometric foundations of possibilistic clustering: A hard possibilistic clustering algorithm","authors":"James C. Bezdek, Thomas A. Runkler","doi":"10.1016/j.fss.2026.109775","DOIUrl":"10.1016/j.fss.2026.109775","url":null,"abstract":"<div><div><em>Possibilistic c-means</em> (PCM) clustering began in 1993, and has been used since then in many applications. In this article we discuss the geometric foundations of PCM and introduce a new <em>hard possibilistic c-means</em> (HPCM) clustering algorithm. We use limit theory to prove that the extended set of possibilistic c-partitions is the unit hypercube in<span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>c</mi><mi>n</mi></mrow></msup></math></span>; and that its vertices are exactly the hard possibilistic c-partitions on n objects defined herein. This enables completion of the geometric description of the domain of possibilistic clustering algorithms. We give examples that compare the results of clustering with <em>Hard c-means</em> (HCM) to HPCM on three small synthetic data sets. Our proof-of-concept examples show that the new algorithm performs as expected, and provides much more realistic interpretation of clusters than HCM when the data contain bridge points or noise.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"532 ","pages":"Article 109775"},"PeriodicalIF":2.7,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146049216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.fss.2026.109770
Kejin Li, Feifei Du
The projective synchronization (PS) of discrete-time fractional-order fuzzy cellular neural networks (DFFCNNs) with distributed delays is investigated in this paper. First, based on the nabla fractional-order difference theory, a comparison principle suitable for fractional-order systems with variable coefficients and multiple time delays is established, and the sub-multiplicative law of the nabla Mittag-Leffler function is rigorously proved. Second, a discrete-time fractional-order Halanay inequality with arbitrary step size, variable coefficients, and multiple time-varying delays is introduced. Furthermore, leveraging the aforementioned inequality, a sufficient condition for the PS of DFFCNNs is derived. Finally, an example is presented to confirm the validity of the results.
{"title":"Exploring projective synchronization in discrete-time fractional-order fuzzy cellular neural networks with distributed delays","authors":"Kejin Li, Feifei Du","doi":"10.1016/j.fss.2026.109770","DOIUrl":"10.1016/j.fss.2026.109770","url":null,"abstract":"<div><div>The projective synchronization (PS) of discrete-time fractional-order fuzzy cellular neural networks (DFFCNNs) with distributed delays is investigated in this paper. First, based on the nabla fractional-order difference theory, a comparison principle suitable for fractional-order systems with variable coefficients and multiple time delays is established, and the sub-multiplicative law of the nabla Mittag-Leffler function is rigorously proved. Second, a discrete-time fractional-order Halanay inequality with arbitrary step size, variable coefficients, and multiple time-varying delays is introduced. Furthermore, leveraging the aforementioned inequality, a sufficient condition for the PS of DFFCNNs is derived. Finally, an example is presented to confirm the validity of the results.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109770"},"PeriodicalIF":2.7,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145963130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.fss.2026.109769
Siyu Xu, Xiaodong Pan, Yexing Dan, Keyun Qin
Recently, by using overlap and grouping functions, Han et al. introduced two novel types of fuzzy rough sets on complete lattices in an L-fuzzy approximation space (U, V, R), along with their application to three-way decisions. We refer to these two types of fuzzy rough sets as the 1st type and the 2nd type of fuzzy rough sets. It is worth noting that, in the case where (U, V, R) is an L-fuzzy approximation space, many properties of these two types of fuzzy rough sets established in L-fuzzy approximation spaces of the form (U, R) (i.e., ) are generally difficult to establish in this more general framework. Therefore, in this paper, we focus on exploring some properties of these two types of fuzzy rough sets under the restriction to an L-fuzzy approximation space (U, R), with particular emphasis on how they generate Alexandrov L-fuzzy topologies. In particular, regarding the 2nd type of fuzzy rough sets, we deduce the behaviours of the upper and lower L-fuzzy rough approximation operators of an L-fuzzy approximation space (U, R) in the case of a family of L-fuzzy relations. Moreover, we explore the relationships among the pair of upper and lower L-fuzzy rough approximation operators proposed by Jiang and Hu in 2022 and the two pairs of upper and lower L-fuzzy rough approximation operators introduced in this study. Our investigations can be regarded as a contribution to enriching the theoretical framework of the two novel types of fuzzy rough sets by Han et al. in an L-fuzzy approximation space (U, V, R).
{"title":"Some properties of two types of fuzzy rough sets on complete lattices constructed by means of overlap and grouping functions","authors":"Siyu Xu, Xiaodong Pan, Yexing Dan, Keyun Qin","doi":"10.1016/j.fss.2026.109769","DOIUrl":"10.1016/j.fss.2026.109769","url":null,"abstract":"<div><div>Recently, by using overlap and grouping functions, Han et al. introduced two novel types of fuzzy rough sets on complete lattices in an <em>L</em>-fuzzy approximation space (<em>U, V, R</em>), along with their application to three-way decisions. We refer to these two types of fuzzy rough sets as the 1<sup><em>st</em></sup> type and the 2<sup><em>nd</em></sup> type of fuzzy rough sets. It is worth noting that, in the case where (<em>U, V, R</em>) is an <em>L</em>-fuzzy approximation space, many properties of these two types of fuzzy rough sets established in <em>L</em>-fuzzy approximation spaces of the form (<em>U, R</em>) (i.e., <span><math><mrow><mi>U</mi><mo>=</mo><mi>V</mi></mrow></math></span>) are generally difficult to establish in this more general framework. Therefore, in this paper, we focus on exploring some properties of these two types of fuzzy rough sets under the restriction to an <em>L</em>-fuzzy approximation space (<em>U, R</em>), with particular emphasis on how they generate Alexandrov <em>L</em>-fuzzy topologies. In particular, regarding the 2<sup><em>nd</em></sup> type of fuzzy rough sets, we deduce the behaviours of the upper and lower <em>L</em>-fuzzy rough approximation operators of an <em>L</em>-fuzzy approximation space (<em>U, R</em>) in the case of a family of <em>L</em>-fuzzy relations. Moreover, we explore the relationships among the pair of upper and lower <em>L</em>-fuzzy rough approximation operators proposed by Jiang and Hu in 2022 and the two pairs of upper and lower <em>L</em>-fuzzy rough approximation operators introduced in this study. Our investigations can be regarded as a contribution to enriching the theoretical framework of the two novel types of fuzzy rough sets by Han et al. in an <em>L</em>-fuzzy approximation space (<em>U, V, R</em>).</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109769"},"PeriodicalIF":2.7,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.fss.2026.109767
Lili Shen, Jian Zhang
Let [0, 1]* be the unit interval [0,1] equipped with a continuous t-norm *. It is shown that the category of [0, 1]*-sets is cartesian closed if, and only if, * is the minimum t-norm on [0,1].
{"title":"Cartesian closedness of the category of real-valued sets, I","authors":"Lili Shen, Jian Zhang","doi":"10.1016/j.fss.2026.109767","DOIUrl":"10.1016/j.fss.2026.109767","url":null,"abstract":"<div><div>Let [0, 1]<sub>*</sub> be the unit interval [0,1] equipped with a continuous t-norm *. It is shown that the category of [0, 1]<sub>*</sub>-sets is cartesian closed if, and only if, * is the minimum t-norm on [0,1].</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109767"},"PeriodicalIF":2.7,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-10DOI: 10.1016/j.fss.2026.109771
Roberto G. Aragón, Jesús Medina, Samuel Molina-Ruiz
In many situations is fundamental to use a procedure to aggregate information obtained from different sources (devices), such as when an edge computing system is used. Bonds were introduced in formal concept analysis as an aggregation method for linking different contexts (datasets) whilst preserving the information they contain. In this paper, we generalize the notion of bond to the multi-adjoint concept lattice framework, which is a fuzzy and flexible extension of formal concept analysis. Furthermore, we study several properties of multi-adjoint bonds defined by the constantly top or constantly bottom relations, with an emphasis on how they aggregate the information in the concept lattices.
{"title":"Context-based sum via multi-adjoint bonds","authors":"Roberto G. Aragón, Jesús Medina, Samuel Molina-Ruiz","doi":"10.1016/j.fss.2026.109771","DOIUrl":"10.1016/j.fss.2026.109771","url":null,"abstract":"<div><div>In many situations is fundamental to use a procedure to aggregate information obtained from different sources (devices), such as when an edge computing system is used. Bonds were introduced in formal concept analysis as an aggregation method for linking different contexts (datasets) whilst preserving the information they contain. In this paper, we generalize the notion of bond to the multi-adjoint concept lattice framework, which is a fuzzy and flexible extension of formal concept analysis. Furthermore, we study several properties of multi-adjoint bonds defined by the constantly top or constantly bottom relations, with an emphasis on how they aggregate the information in the concept lattices.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109771"},"PeriodicalIF":2.7,"publicationDate":"2026-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-08DOI: 10.1016/j.fss.2026.109766
Zhenyu Xiu , Xu Zheng
In this paper, we primarily investigate methods for generating uninorms on complete lattices using monotone functions and their pseudo-inverses. First, we present a construction of a uninorm on a complete lattice based on a t-norm, a complete inf-homomorphism, and its pseudo-inverse. Next, we introduce a new method for generating a uninorm via a given uninorm, a complete inf-homomorphism, and its pseudo-inverse. Finally, we explore methods for constructing a uninorm on a complete lattice using a given uninorm together with an injective complete inf-homomorphism and its pseudo-inverse.
{"title":"Construction of uninorms on complete lattices by a monotone function and its pseudo-inverse","authors":"Zhenyu Xiu , Xu Zheng","doi":"10.1016/j.fss.2026.109766","DOIUrl":"10.1016/j.fss.2026.109766","url":null,"abstract":"<div><div>In this paper, we primarily investigate methods for generating uninorms on complete lattices using monotone functions and their pseudo-inverses. First, we present a construction of a uninorm on a complete lattice based on a t-norm, a complete inf-homomorphism, and its pseudo-inverse. Next, we introduce a new method for generating a uninorm via a given uninorm, a complete inf-homomorphism, and its pseudo-inverse. Finally, we explore methods for constructing a uninorm on a complete lattice using a given uninorm together with an injective complete inf-homomorphism and its pseudo-inverse.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109766"},"PeriodicalIF":2.7,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-08DOI: 10.1016/j.fss.2026.109762
Qian Wang , Lian Duan , Lihong Huang , Xianwen Fang
It is known that finite-time synchronization is crucial in engineering applications, such as blockchain consensus protocols. Additionally, fields such as smart grids, UAV swarms, and autonomous driving rely on finite-time synchronization to enhance the robustness and real-time performance of cooperative control, preventing performance degradation or safety risks caused by communication delays. This paper is concerned with the finite-time synchronization problem of fuzzy delayed Cohen-Grossberg neural networks (CGNNs) in which the activation functions are discontinuous. By designing delay-independent feedback controllers combined with new analytical techniques, some novel finite-time synchronization criteria are established without using the widely employed finite-time stability theory, which greatly enriches the theory of complex neurodynamics. Finally, a numerical example is provided to illustrate the effectiveness of the theoretical results.
{"title":"New results on finite-time synchronization of fuzzy delayed Cohen-Grossberg neural networks with discontinuous activations","authors":"Qian Wang , Lian Duan , Lihong Huang , Xianwen Fang","doi":"10.1016/j.fss.2026.109762","DOIUrl":"10.1016/j.fss.2026.109762","url":null,"abstract":"<div><div>It is known that finite-time synchronization is crucial in engineering applications, such as blockchain consensus protocols. Additionally, fields such as smart grids, UAV swarms, and autonomous driving rely on finite-time synchronization to enhance the robustness and real-time performance of cooperative control, preventing performance degradation or safety risks caused by communication delays. This paper is concerned with the finite-time synchronization problem of fuzzy delayed Cohen-Grossberg neural networks (CGNNs) in which the activation functions are discontinuous. By designing delay-independent feedback controllers combined with new analytical techniques, some novel finite-time synchronization criteria are established without using the widely employed finite-time stability theory, which greatly enriches the theory of complex neurodynamics. Finally, a numerical example is provided to illustrate the effectiveness of the theoretical results.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"530 ","pages":"Article 109762"},"PeriodicalIF":2.7,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-07DOI: 10.1016/j.fss.2026.109761
Pavel Martinek
The paper provides a survey of several ways how to describe fuzzy multiset regular languages, i.e., languages generated by fuzzy multiset regular grammars. These languages can also be characterized by means of fuzzy multiset finite automata (both in general and in reduced forms), fuzzy multiset regular expressions, and as fuzzy multiset languages which can be expressed in a semilinear form. Moreover, it is pointed out that a prevailing number of already published papers concerning fuzzy multiset finite automata is based on a wrong definition. It is also shown that the name ‘deterministic fuzzy multiset finite automaton’ is often used incorrectly for automata deserving adjective pseudodeterministic.
{"title":"Fuzzy multiset regular languages and their basic characterizations","authors":"Pavel Martinek","doi":"10.1016/j.fss.2026.109761","DOIUrl":"10.1016/j.fss.2026.109761","url":null,"abstract":"<div><div>The paper provides a survey of several ways how to describe fuzzy multiset regular languages, i.e., languages generated by fuzzy multiset regular grammars. These languages can also be characterized by means of fuzzy multiset finite automata (both in general and in reduced forms), fuzzy multiset regular expressions, and as fuzzy multiset languages which can be expressed in a semilinear form. Moreover, it is pointed out that a prevailing number of already published papers concerning fuzzy multiset finite automata is based on a wrong definition. It is also shown that the name ‘deterministic fuzzy multiset finite automaton’ is often used incorrectly for automata deserving adjective pseudodeterministic.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109761"},"PeriodicalIF":2.7,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Long-term time series forecasting constitutes a significant area of research within data mining, pattern analysis, and recognition. Although the Gaussian linear fuzzy information granule (GLFIG) has garnered increasing attention as an efficient approach for long-term time series forecasting, it continues to face substantial challenges, including the quantification of trend extraction objectives, the non-metric nature of granular distance formulations, and the effective mining of granular structural information. To address these challenges, this study designs a long-term forecasting model based on GLFIGs, incorporating optimized trend information and enhanced periodic patterns. First, the model improves the extraction of trend information through an l1-trend filter, which provides a principled basis for parameter selection. Second, periodic structures within the granular time series are refined to mitigate structural distortions caused by real-world data complexity, and a metric-compliant distance formula for GLFIGs of unequal length is introduced for the first time. Finally, a CNN-LSTM architecture augmented with periodic information is employed for long-term forecasting, leveraging long short-term memory (LSTM) to complement the limited temporal sensitivity of convolutional neural networks (CNNs). Experiments conducted on ten publicly available time series datasets demonstrate that the proposed model achieves satisfactory predictive accuracy in long-term univariate time series forecasting.
{"title":"A long-term prediction model with Gaussian linear fuzzy granules based on convolutional neural networks and long short-term memory","authors":"Xueling Ma , Chenglong Zhu , Weiping Ding , Pierpaolo D’ Urso , Jianming Zhan","doi":"10.1016/j.fss.2026.109765","DOIUrl":"10.1016/j.fss.2026.109765","url":null,"abstract":"<div><div>Long-term time series forecasting constitutes a significant area of research within data mining, pattern analysis, and recognition. Although the Gaussian linear fuzzy information granule (GLFIG) has garnered increasing attention as an efficient approach for long-term time series forecasting, it continues to face substantial challenges, including the quantification of trend extraction objectives, the non-metric nature of granular distance formulations, and the effective mining of granular structural information. To address these challenges, this study designs a long-term forecasting model based on GLFIGs, incorporating optimized trend information and enhanced periodic patterns. First, the model improves the extraction of trend information through an <em>l</em><sub>1</sub>-trend filter, which provides a principled basis for parameter selection. Second, periodic structures within the granular time series are refined to mitigate structural distortions caused by real-world data complexity, and a metric-compliant distance formula for GLFIGs of unequal length is introduced for the first time. Finally, a CNN-LSTM architecture augmented with periodic information is employed for long-term forecasting, leveraging long short-term memory (LSTM) to complement the limited temporal sensitivity of convolutional neural networks (CNNs). Experiments conducted on ten publicly available time series datasets demonstrate that the proposed model achieves satisfactory predictive accuracy in long-term univariate time series forecasting.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"530 ","pages":"Article 109765"},"PeriodicalIF":2.7,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-06DOI: 10.1016/j.fss.2026.109764
Yao Ouyang , Qiuyu Cao , Hua-Peng Zhang
Given a curvilinear section, there are three known copulas. We prove that two of them are always singular and the third one is also singular under some mild condition. With the aid of the two always singular copulas and by employing the rectangular patchwork of copulas, we explore when there exists an absolutely continuous copula with a given curvilinear section. Several sufficient conditions and one necessary condition are discovered for the existence problem. The Durante-Jaworski theorem is retrieved when the curvilinear section reduces to a diagonal section.
{"title":"Absolutely continuous copulas with a given curvilinear section","authors":"Yao Ouyang , Qiuyu Cao , Hua-Peng Zhang","doi":"10.1016/j.fss.2026.109764","DOIUrl":"10.1016/j.fss.2026.109764","url":null,"abstract":"<div><div>Given a curvilinear section, there are three known copulas. We prove that two of them are always singular and the third one is also singular under some mild condition. With the aid of the two always singular copulas and by employing the rectangular patchwork of copulas, we explore when there exists an absolutely continuous copula with a given curvilinear section. Several sufficient conditions and one necessary condition are discovered for the existence problem. The Durante-Jaworski theorem is retrieved when the curvilinear section reduces to a diagonal section.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"530 ","pages":"Article 109764"},"PeriodicalIF":2.7,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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